Elvira Nabiullina has earned an unusual degree of freedom to buttress an economy buffeted by sanctions

ST. PETERSBURG—After Russia’s central-bank chief, Elvira Nabiullina, moved to shut down a large lender last year for allegedly falsifying accounts, the nation’s top prosecutor’s office issued an order to leave the bank alone.

She closed it anyway.

In her five years in office, Ms. Nabiullina has closed hundreds of weak banks, stymied the exodus...